Let (Ӽ, ᶁ_1) and (Ƴ,ᶁ_2) be compact metric spaces, ʄ:(Ӽ,ᶁ_1) →(Ӽ,ᶁ_1) and : (Ƴ,ᶁ_2) →(Ƴ,ᶁ_2) be continuous maps. If ʄ and G have dense minimal points and the average inverse shadowing property, we have proved ʄ×G has an average inverse shadowing property, topological transitive and dense minimal points. Moreover, we have proved ʄ is totally strongly ergodic and weakly mixing.